NONMONOTONE IMPULSE EFFECTS IN SECOND-ORDER PERIODIC BOUNDARY VALUE PROBLEMS

We deal with the nonlinear impulsive periodic boundary value problem u′′ = f (t,u,u′), u(ti+)= Ji(u(ti)), u′(ti+)=Mi(u′(ti)), i= 1,2, . . . ,m, u(0)= u(T), u′(0)= u′(T). We establish the existence results which rely on the presence of a well-ordered pair (σ1,σ2) of lower/upper functions (σ1 ≤ σ2 on [0,T]) associated with the problem. In contrast to previous papers investigating such problems, the monotonicity of the impulse functions Ji, Mi is not required here.


Introduction
In recent years, the theory of impulsive differential equations has become a well-respected branch of mathematics.This is because of its characteristic features which provide many interesting problems that cannot be solved by applying standard methods from the theory of ordinary differential equations.It can also give a natural description of many real models from applied sciences (see the examples mentioned in [1,2]).
In particular, starting with [7], periodic boundary value problems for nonlinear second-order impulsive differential equations of the form (2.1), (2.2), and (2.3) have received considerable attention; see, e.g., [1,3,5,6,8,9,14], where the existence results in terms of lower and upper functions can also be found.However, all impose certain monotonicity requirements on the impulse functions.In contrast to these papers, we provide existence results using weaker conditions (2.10) and (2.11) instead of monotonicity.
Throughout the paper, we keep the following notation and conventions.For a real valued function u defined a.e. on [0,T], we put For a given interval J ⊂ R, let C(J) denote the set of real-valued functions which are continuous on J. Furthermore, let C 1 (J) be the set of functions having continuous first derivatives on J, and L(J) the set of functions which are Lebesgue integrable on J.
578 Nonmonotone impulse effects in periodic problems Let m ∈ N and let be a division of the interval [0,T].We denote D = t 1 ,t 2 ,...,t m (1.3) and define C 1 D [0,T] as the set of functions u : [0,T] → R, where having first derivatives absolutely continuous on each subinterval (t i ,t i+1 ), i = 0,1,...,m.For u ∈ C 1 D [0,T] and i = 1,2,...,m + 1, we write ) Note that the set C 1 D [0,T] becomes a Banach space when equipped with the norm • D and with the usual algebraic operations.
We say that f : holds for a.e.t ∈ [0,T] and all (x, y) ∈ K.The set of functions satisfying the Carathéodory conditions on [0, T] × R 2 will be denoted by Car([0,T] × R 2 ).
Given a Banach space X and its subset M, let cl(M) and ∂M denote the closure and the boundary of M, respectively.
Let Ω be an open bounded subset of X. Assume that the operator F : cl(Ω) → X is completely continuous and F u = u for all u ∈ ∂Ω.Then deg(I − F,Ω) denotes the Leray-Schauder topological degree of I − F with respect to Ω, where I is the identity operator on X.For a definition and properties of the degree, see, for example, [4] or [10].

Formulation of the problem and main assumptions
Here we study the existence of solutions to the following problem: (2.1) ) where u (t i ) are understood in the sense of (1.5), f ∈ Car([0,T] × R 2 ), J i ∈ C(R), and M i ∈ C(R).

A priori estimates
At the beginning of this section, we introduce a class of auxiliary problems and prove uniform a priori estimates for their solutions.Take and all and all (3.1) and consider an auxiliary Dirichlet problem ) (2.9), and (2.10) and (3.1), (3.2), (3.3), and (3.4) hold.Then every solution u of (3.5), (3.6), and (3.7) satisfies Proof.Let u be a solution of (3.5), (3.6), and Then, by (3.4), we have So, it remains to prove that v ≤ 0 on (0,T).
To summarize, we have proved that v ≤ 0 on [0,T] which means that u ≤ σ 2 on [0, T].If we put v = σ 1 − u on [0, T] and use the properties of σ 1 instead of σ 2 , we can prove σ 1 ≤ u on [0, T] by an analogous argument.
In the proof of Theorem 4.1, we need a priori estimates for derivatives of solutions.To this aim we prove the following lemma.Lemma 3.2.Assume that r ∈ (0,∞) and that ) (3.25) Then there exists r * ∈ (1,∞) such that the estimate Before proving this theorem, we prove the next key proposition where we restrict ourselves to the case that f is bounded by a Lebesgue integrable function.

Proof
Step 1.We construct a proper auxiliary problem.
Step 3. We find estimates for solutions of the auxiliary problem.